Sigma function

Equation (3) has good quantitative predictive power and is a successful extra-thermodynamic relationship like the Hammett sigma function. No other approach to modeling complex formation equilibria, including HSAB itself, can predict log values for unidentate ligands so accurately. 

These two systems are examples from non-linear physics, where the equations can be solved in terms of elliptic functions and elliptic integrals. The reader who is not familiar with these functions, which do not arise in the same way as the previously mentioned special functions, is referred to the excellent book by Whittaker and Watson [6]. In that book, the reader will see that there are two flavours of elliptic functions, the Weierstrass and Jacobi representations, three kinds of elliptic integrals, and six kinds of pseudo-periodic functions, the Weierstrass zeta and sigma functions and the four kinds of Jacobi theta functions. Of historical interest for theoretical chemists is the fact that Jacobi s imaginary transformation of the theta functions is the same as the Ewald transformation of crystal physics [7]. 

Correlations of with Other Properties. With this introduction to the measurement of we shall examine the important relationships which have been found between the physical properties of H bonded systems and the shift of brought about by the H bond interaction. Some of these relationships are summarized qualitatively in Fig. 3-6. These values of Av refer to a variety of systems, each with a well defined H bonded molecular species (in contrast to the Avs data of Fig. 3-5, which refer to the superposition of absorptions of many different polymeric species of a single type of AB molecules). In Fig. 3-6 we see that Av is inversely related to the A 3 distance (/ ), and Hammett sigma function, whereas it is directly related to enthalpy of H bond formation, the A—H distance (r), and the two IR spectral properties, half-width and intensity. 

Figure 11. Correlation of triboelectric charging with electronegativity, as measured by Hammet sigma function. (Reproduced from Ref. 66. Copyright 1975, American Chemical Society.)...

Hammett s sigma function as revised by Taft (Taft s o ). ... 

Hertel, J. (2009). Computing the uncomputable rado sigma function An automated, symbolic induction prover for nonhalting Turing machines. The Mathematica Journal, 11(2), 270-283. Available from http //www.mathematica-journal. com/issue/vl 112/contents/Hertel/Hertel.pdf... 

Taft sigma values cr perform a similar function with respect to aliphatic and alicyclic systems. Values of cr are listed in Table 9.1. 

The current functional model of o control is depicted in Fig. 2. In the absence of non-native proteins in the periplasm or outer membrane, o is sequestered by RseA acting as an anti-sigma factor. Tight binding of o requires the participation of RseB which might act as an co-anti-sigma factor. Upon accumulation of non-native proteins outside the cytoplasm, RseB is released from... 

In addition to these actions, glucocorticoids modify the function of several transmitters and/or their receptors, notably GABA, acetylcholine, noradrenaline, 5-HT and sigma (cr)-receptors for which the endogenous ligand is unknown. Conversely, the... 

M 0004. (From Tiffany and Krimm, 1969, Biopolymers 8, 347-359, 1969. Reprinted by permission of John Wiley Sons, Inc.) (B) CD spectrum of poly-L-proline II (sigma, molecular weight 55,000) as a function of temperature. (C) CD spectrum of polyglutamic acid at pH 7 as a function of temperature. (D) CD spectrum of poly-L-lysine at pH 7 as a function of temperature. Parts (B-D) are from Tiffany and Krimm (1972). Biopolymers 11, 2309-2316, 1972. Reprinted by permission of John Wiley 8c Sons, Inc. 

Figure 43-6 Family of curves of multiplication factor as a function of Er, for different values of the parameter sigma (the noise standard deviation), for Normally distributed error. Values of sigma range from 0.1 to 1.0 for the ten curves shown, (see Color Plate 5)...

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